Align UV map island with axis or other island
Change the snapping from increment to vertex 

Select an island with L
Choose one of the vertices on the island and match it to the corresponding one on the other island



If they are not aligned due to rotation, continue with these steps


Select the vertex that matches the island you are aligning to. 
Use Shift-S to snap the cursor to the selected vertex
Select the one island with L
Use . to set the pivot point to be the cursor
Rotate the island until R and snap a vertex that isn't lined up with the cursor to the similar one on the other mesh.



If the scale doesn't match proceed further these steps


With the pivot point still at the cursor at the matching vertex, hit S to scale until it snaps into place


If the map is distorted, or something above didn't work right, do these procedures instead of the above steps


Snap each vertex individually to the corresponding vertex using G.



Note: if the snapping doesn't work, try holding down Ctrl